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Difference between revisions of "2022 AMC 10A Problems/Problem 25"

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==Solution==
 
==Solution==
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Let <math>r</math> be the number of lattice points on the side length of square <math>R</math>, <math>s</math> be the number of lattice points on the side length of square <math>S</math>, and <math>t</math> be the number of lattice points on the side length of square <math>T</math>. Note that the actual lengths of the side lengths are the number of lattice points minus <math>1</math>, so we can work in terms of <math>r, s, t</math> and subtract <math>3</math> to get the actual answer at the end. Furthermore, note that the number of lattice points inside a rectangular region is equal to the number of lattice points in its width times the number of lattice points along its length.
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Using this fact, the number of lattice points in <math>R</math> is <math>r^2</math>, the number of lattice points in <math>S</math> is <math>s^2</math>, and the number of lattice points in <math>T</math> is <math>t^2</math>.
 
Solution in progress
 
Solution in progress
  

Revision as of 23:38, 12 November 2022

Problem 25

Let $R$, $S$, and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the x-axis. The left edge of $R$ and the right edge of $S$ are on the $y$-axis, and $R$ contains $\frac{9}{4}$ as many lattice points as does $S$. The top two vertices of $T$ are in $R \cup S$, and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S$. See the figure (not drawn to scale).

P252022A.png

The fraction of lattice points in $S$ that are in $S \cap T$ is 27 times the fraction of lattice points in $R$ that are in $R \cap T$. What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$?

$\textbf{(A) }336\qquad\textbf{(B) }337\qquad\textbf{(C) }338\qquad\textbf{(D) }339\qquad\textbf{(E) }340$

Solution

Let $r$ be the number of lattice points on the side length of square $R$, $s$ be the number of lattice points on the side length of square $S$, and $t$ be the number of lattice points on the side length of square $T$. Note that the actual lengths of the side lengths are the number of lattice points minus $1$, so we can work in terms of $r, s, t$ and subtract $3$ to get the actual answer at the end. Furthermore, note that the number of lattice points inside a rectangular region is equal to the number of lattice points in its width times the number of lattice points along its length.

Using this fact, the number of lattice points in $R$ is $r^2$, the number of lattice points in $S$ is $s^2$, and the number of lattice points in $T$ is $t^2$. Solution in progress

~KingRavi

See Also

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
[[2022 AMC 10A Problems/Problem {{{num-a}}}|Problem {{{num-a}}}]]
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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